Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/202

CIRCLE OF CURVATURE. CENTER OF CURVATURE

116. Circle of curvature.^[1] Center of curvature. If a circle be drawn through three points P₀, P₁, P₂ on a plane curve, and if P₁ and P₂ be made to approach P₀ along the curve as a limiting position, then the circle will in general approach in magnitude and position a limiting circle called the circle of curvature of the curve at the point P₀. The center of this circle is called the center of curvature.

Let the equation of the curve be

(1)	$y\ =\ f(x)$;

and let $x_{0},x_{1},x_{2}$ be the abscissas of the points $P_{0},P_{1},P_{2}$ respectively, $(\alpha ',\beta ')$ the coördinates of the center, and $R'$ the radius of the circle passing through the three points. Then the equation of the circle is

$(x-\alpha ')^{2}+(y-\beta ')^{2}\ =\ R'{^{2}};$

and since the coordinates of the points $P_{0},P_{1},P_{2}$ must satisfy this equation, we have

(2)	${\begin{cases}(x_{0}-\alpha ')^{2}+(y_{0}-\beta ')^{2}-R'^{2}=0,\\(x_{1}-\alpha ')^{2}+(y_{1}-\beta ')^{2}-R'^{2}=0,\\(x_{2}-\alpha ')^{2}+(y_{2}-\beta ')^{2}-R'^{2}=0.\end{cases}}$

Now consider the function of $X$ defined by

$F(x)\ =\ (x-\alpha ')^{2}+(y-\beta ')^{2}=R'2,$

in which $y$ has been replaced by $f(x)$ from (1).

Then from equations (2) we get

$F(x_{0})\ =\ 0,F(x_{1})\ =\ 0,F(x_{2})\ =\ 0.$

1. Sometimes called the osculating circle. The circle of curvature was defined from another point of view on §104